↑

Properties

Label 2-197-1.1-c9-0-58
Degree 22
Conductor 197197
Sign 11
Analytic cond. 101.462101.462
Root an. cond. 10.072810.0728
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 28.5·2-s − 17.8·3-s + 303.·4-s − 2.27e3·5-s + 510.·6-s + 4.44e3·7-s + 5.94e3·8-s − 1.93e4·9-s + 6.50e4·10-s + 8.30e4·11-s − 5.43e3·12-s + 1.87e5·13-s − 1.27e5·14-s + 4.07e4·15-s − 3.25e5·16-s + 6.67e5·17-s + 5.53e5·18-s + 4.11e5·19-s − 6.91e5·20-s − 7.95e4·21-s − 2.37e6·22-s − 7.75e5·23-s − 1.06e5·24-s + 3.23e6·25-s − 5.36e6·26-s + 6.98e5·27-s + 1.35e6·28-s + ⋯
L(s)  = 1  − 1.26·2-s − 0.127·3-s + 0.593·4-s − 1.62·5-s + 0.160·6-s + 0.700·7-s + 0.513·8-s − 0.983·9-s + 2.05·10-s + 1.71·11-s − 0.0755·12-s + 1.82·13-s − 0.884·14-s + 0.207·15-s − 1.24·16-s + 1.93·17-s + 1.24·18-s + 0.724·19-s − 0.966·20-s − 0.0892·21-s − 2.15·22-s − 0.577·23-s − 0.0654·24-s + 1.65·25-s − 2.30·26-s + 0.252·27-s + 0.415·28-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 11
Analytic conductor: 101.462101.462
Root analytic conductor: 10.072810.0728
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 197, ( :9/2), 1)(2,\ 197,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 1.0866300691.086630069
L(12)L(\frac12) ≈\approx 1.0866300691.086630069
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad197 1−1.50e9T 1 - 1.50e9T
good2 1+28.5T+512T2 1 + 28.5T + 512T^{2}
3 1+17.8T+1.96e4T2 1 + 17.8T + 1.96e4T^{2}
5 1+2.27e3T+1.95e6T2 1 + 2.27e3T + 1.95e6T^{2}
7 1−4.44e3T+4.03e7T2 1 - 4.44e3T + 4.03e7T^{2}
11 1−8.30e4T+2.35e9T2 1 - 8.30e4T + 2.35e9T^{2}
13 1−1.87e5T+1.06e10T2 1 - 1.87e5T + 1.06e10T^{2}
17 1−6.67e5T+1.18e11T2 1 - 6.67e5T + 1.18e11T^{2}
19 1−4.11e5T+3.22e11T2 1 - 4.11e5T + 3.22e11T^{2}
23 1+7.75e5T+1.80e12T2 1 + 7.75e5T + 1.80e12T^{2}
29 1+3.43e6T+1.45e13T2 1 + 3.43e6T + 1.45e13T^{2}
31 1−8.79e6T+2.64e13T2 1 - 8.79e6T + 2.64e13T^{2}
37 1+8.14e5T+1.29e14T2 1 + 8.14e5T + 1.29e14T^{2}
41 1−4.73e6T+3.27e14T2 1 - 4.73e6T + 3.27e14T^{2}
43 1−3.02e7T+5.02e14T2 1 - 3.02e7T + 5.02e14T^{2}
47 1−2.02e7T+1.11e15T2 1 - 2.02e7T + 1.11e15T^{2}
53 1+3.68e7T+3.29e15T2 1 + 3.68e7T + 3.29e15T^{2}
59 1−1.23e8T+8.66e15T2 1 - 1.23e8T + 8.66e15T^{2}
61 1−9.31e7T+1.16e16T2 1 - 9.31e7T + 1.16e16T^{2}
67 1−3.56e7T+2.72e16T2 1 - 3.56e7T + 2.72e16T^{2}
71 1−4.86e7T+4.58e16T2 1 - 4.86e7T + 4.58e16T^{2}
73 1+1.24e8T+5.88e16T2 1 + 1.24e8T + 5.88e16T^{2}
79 1+1.55e8T+1.19e17T2 1 + 1.55e8T + 1.19e17T^{2}
83 1+6.28e8T+1.86e17T2 1 + 6.28e8T + 1.86e17T^{2}
89 1+1.00e8T+3.50e17T2 1 + 1.00e8T + 3.50e17T^{2}
97 1+1.05e9T+7.60e17T2 1 + 1.05e9T + 7.60e17T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.05485799354854337566997207586, −9.658112958623149784499351888057, −8.488549708611254426710376498850, −8.232093219561504759219735705175, −7.24026632112568866365621543623, −5.86570836986732187889687085464, −4.21819392318863602708663629787, −3.42444842162520505846402605313, −1.23941678130265676368181864720, −0.77043531220448362747622950414, 0.77043531220448362747622950414, 1.23941678130265676368181864720, 3.42444842162520505846402605313, 4.21819392318863602708663629787, 5.86570836986732187889687085464, 7.24026632112568866365621543623, 8.232093219561504759219735705175, 8.488549708611254426710376498850, 9.658112958623149784499351888057, 11.05485799354854337566997207586

Graph of the ZZ-function along the critical line